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41
From Heisenberg to Gödel via Chaitin
, 2008
"... In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa”. Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. A ..."
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Cited by 11 (9 self)
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In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa”. Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a “formal uncertainty principle ” which implies Chaitin’s informationtheoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. This fact supports the conjecture that uncertainty implies randomness not only in mathematics, but also in physics.
The growth of mathematical knowledge: an open world view
 The growth of mathematical knowledge, Kluwer, Dordrecht 2000
"... mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but ..."
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Cited by 5 (5 self)
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mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past ” (Poincaré 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself ” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks
Lifestyles  A Paradigm for the description of spatiotemporal databases
 Department of Geoinformation. Vienna, Technical University
, 1999
"... This thesis investigates operations affecting identity of objects in a spatiotemporal database, ubiquitous for future temporal geographic information systems (GIS). Two different techniques to record change in temporal databases are compared: database versioning and object versioning. We show formal ..."
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Cited by 4 (0 self)
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This thesis investigates operations affecting identity of objects in a spatiotemporal database, ubiquitous for future temporal geographic information systems (GIS). Two different techniques to record change in temporal databases are compared: database versioning and object versioning. We show formally that these techniques are equivalent and use conceptually simpler model of database versioning for the further development. The conceptual model of our database is based on the entityrelationship model. The complete temporal database is an appendonly series of snapshots, each of which represents the state of the universe of discourse at a particular moment on the time scale. Each snapshot consists of a set of objects connected with relations. Objects are metaphorically perceived as having life: an object has its birth or creation, its life or existence, its death or destruction. The central concept in the life of an object is its identifier, which is unchanged from the birth to the death
Truth and Light: Physical Algorithmic Randomness
, 2005
"... This thesis examines some problems related to Chaitin's Ω number. In the first section, we describe several new minimalist prefixfree machines suitable for the study of concrete algorithmic information theory; the halting probabilities of these machines are all Ω numbers. In the second part, ..."
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Cited by 3 (2 self)
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This thesis examines some problems related to Chaitin's Ω number. In the first section, we describe several new minimalist prefixfree machines suitable for the study of concrete algorithmic information theory; the halting probabilities of these machines are all Ω numbers. In the second part, we show that when such a sequence is the result given by a measurement of a system, the system itself can be shown to satisfy an uncertainty principle equivalent to Heisenberg's uncertainty principle. This uncertainty principle also implies Chaitin's strongest form of incompleteness. In the last part, we show that Ω can be written as an infinite product over halting programs; that there exists a "natural," or basefree formulation that does not (directly) depend on the alphabet of the universal prefixfree machine; that Tadaki's generalized halting probability is welldefined even for arbitrary univeral Turing machines and the plain complexity; and finally, that the natural generalized halting probability can be written as an infinite product over primes and has the form of a zeta function whose zeros encode halting information. We conclude with some speculation about physical systems in which partial randomness could arise, and identify many open problems.
Formal Methods in VLSI System Design
, 1996
"... We apply mathematical logic to a number of problems arising in very large scale integration (VLSI) design automation. The first stage of this dissertation is concerned with techniques for the efficient verification of digital systems. We introduce heuristics based on Binary Decision Diagrams for eff ..."
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Cited by 3 (1 self)
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We apply mathematical logic to a number of problems arising in very large scale integration (VLSI) design automation. The first stage of this dissertation is concerned with techniques for the efficient verification of digital systems. We introduce heuristics based on Binary Decision Diagrams for efficiently representing designs specified as gatelevel circuits. We also present an approach to verifying hierarchical designs which uses novel notions of state equivalence to simplify components. The second stage addresses the problem of synthesizing digital designs. We use the logic S1S to demonstrate that the flexibility available for optimizing components in hierarchical designs can be characterized by a finite state automaton. This approach is extended to the problem of synthesizing p...
What does it mean to say that logic is formal
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this
1998] ”Perceiving hierarchical structure in nonrepresentational paintings
 Empirical Studies of the Arts
"... � This paper was originally published in Empirical Studies of the Arts. Vol. 16(1) ..."
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Cited by 2 (1 self)
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� This paper was originally published in Empirical Studies of the Arts. Vol. 16(1)
Semantical Investigation of Simultaneous Skolemization for FirstOrder Sequent Calculus
, 1998
"... Simultaneous quantifier elimination in sequent calculus is an improvement over the wellknown skolemization. It allows a lazy handling of instantiations as well as of the order of certain reductions. We prove the soundness of a sequent calculus which incorporates a rule for simultaneous quantifier e ..."
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Cited by 1 (1 self)
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Simultaneous quantifier elimination in sequent calculus is an improvement over the wellknown skolemization. It allows a lazy handling of instantiations as well as of the order of certain reductions. We prove the soundness of a sequent calculus which incorporates a rule for simultaneous quantifier elimination. The proof is performed by semantical arguments and provides some insights into the dependencies between various formulas in a sequent.